Design of FIR filters with complex desired frequency response using a generalized Remez algorithm

Abstract

Absfract-Complex approximation with a generalized Remez algorithm is used to design FIR digital filters with nonconjugate symmetric frequency responses. The minimax criterion is used and the Chebychev approximation is posed as a linear optimization problem. The primal problem is converted to its dual and is solved using an efficient, quadratically convergent algorithm developed by Tang [l]. Optimal Chebychev real-coefficient FIR filters with group delay smaller than half the filter length can be designed with slightly better magnitude responses compared to linear-phase filters. Linear-phase filters can also be designed when the group delay is specified to be half the filter length. Most importantly, the design method is capable to produce filters with complex coefficients that approximate noncoqjugate symmetric frequency responses.

Complex Chebyshev approximation for FIR filters using linear semi‐infinite programming method

“…As described in Section 3, if problem (12) is correctly solved, it is possible to find the correct number of frequency points of the optima from the number of effective constraints. However, since the selection operation consists of simply solving approximation problem (12), the solution of ω for which the constraints are effective contains neighborhood points of the true frequency points of the optima. In particular, if the frequency points of the optima are degenerate, there are cases in which more than two adjacent constraints can (11) (10) (9) (12) (13) become effective simultaneously near the true frequency points of optima.…”

“…In the adjustment process, let the positions of the effective constraints be (ω m , t m ), while y(ω m , t m ) and x′ are the initial values. The adjustment is made such that the Karush-Kuhn-Tucker condition, the optimization condition for the solution to problem (12), is satisfied. Here, x′ is the solution to the principal problem derived from y(ω m , t m ) and (ω m , t m ) by means of the Complementary Slack Theorem [22].…”

“…In the selection operation, candidate effective constraints for problem (12) are selected. To this end, ω is divided into L segments on Ω and t is divided into P segments on T, so that problem (12) is reduced to the following linear programming problem with finite constraints:…”

Design of FIR Hilbert transformers and differentiators in the complex domain

“…Recently, a new method has been presented for the design of FIR filters in the complex domain [1]. The method can approximate a frequency response with specification of both the phase and magnitude functions.…”

On the Applications of the Frequency-Response Masking Technique in Array Beamforming

“…This technique has become the method of choice primarily because of its considerable complexity reduction in implemen- tation compared to other FIR filter design alternatives, such as those in [6], [7], [19]. Further complexity reduction has been an active topic of research recently [8]- [10] and is one of the objectives of this special issue.…”